A069466: Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.

A069466 palindromic γ-positive nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing · Hadamard kernel what this means

Recurrence from OEIS A069466

\(\left(n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x - 8\,n - 8\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-12 + 24\,x - 12\,x^{2} + 32\,n - 64\,n x + 32\,n x^{2} - 16\,n^{2} + 32\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	6	24	6													36
3	20	180	180	20												400
4	70	1120	2520	1120	70											4900
5	252	6300	25200	25200	6300	252										63504
6	924	33264	207900	369600	207900	33264	924									853776
7	3432	168168	1513512	4204200	4204200	1513512	168168	3432								11778624
8	12870	823680	10090080	40360320	63063000	40360320	10090080	823680	12870							165636900
9	48620	3938220	63011520	343062720	771891120	771891120	343062720	63011520	3938220	48620						2363904400
10	184756	18475600	374130900	2660486400	8147739600	11732745024	8147739600	2660486400	374130900	18475600	184756					34134779536
11	705432	85357272	2133931800	19205386200	76821544800	150570227808	150570227808	76821544800	19205386200	2133931800	85357272	705432				497634306624
12	2704156	389398464	11779303536	130881150400	662585823900	1696219709184	2308743493056	1696219709184	662585823900	130881150400	11779303536	389398464	2704156			7312459672336
13	10400600	1757701400	63277250400	850727477600	5317046735000	17227231421400	30626189193600	30626189193600	17227231421400	5317046735000	850727477600	63277250400	1757701400	10400600		108172480360000
14	40116600	7862853600	332205564600	5315289033600	40196873316600	160787493266400	361771859849400	472518347558400	361771859849400	160787493266400	40196873316600	5315289033600	332205564600	7862853600	40116600	1609341595560000

Generated b-files: A002894 (row sums), A069466, A069466 (reversed triangle) ?

Computed in your browser ?

Technical details

Node id: n-fdc03fad975441d0 · revision 1 · JSON
Definition source: oeis_catalog:A069466
Fingerprint: f6-b8a804edad022635

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A069466: reversed triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A069466: triangle, 45 terms agree, 9 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?
A002894: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A069466: triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

The row polynomials satisfy n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A069466: reversed triangle, offset 0, from the row P₀

%e Example

Triangle begins:
  0:     1;
  1:     2,       2;
  2:     6,      24,        6;
  3:    20,     180,      180,        20;
  4:    70,    1120,     2520,      1120,        70;
  5:   252,    6300,    25200,     25200,      6300,       252;
  6:   924,   33264,   207900,    369600,    207900,     33264,       924;
  7:  3432,  168168,  1513512,   4204200,   4204200,   1513512,    168168,     3432;
  8: 12870,  823680, 10090080,  40360320,  63063000,  40360320,  10090080,   823680,   12870;
  9: 48620, 3938220, 63011520, 343062720, 771891120, 771891120, 343062720, 63011520, 3938220, 48620;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n^2*P(n,x) = (8*n^2-8*n+2+(8*n^2-8*n+2)*x)*P(n-1,x) - (16*n^2-32*n+12-(32*n^2-64*n+24)*x+(16*n^2-32*n+12)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*P[n - 2, x])/(n^2)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 8*n + 2 + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (16*n^2 - 32*n + 12 - (32*n^2 - 64*n + 24)*x + (16*n^2 - 32*n + 12)*x^2)*v[i - 2])/(n^2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-412107fab2ca179a
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-7dc194e46d1f26fe
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-7dc194e46d1f26fe
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-7dc194e46d1f26fe
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-7dc194e46d1f26fe

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

theorem_check three_term_recurrence e-412107fab2ca179a root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
theorem_check three_term_recurrence e-7dc194e46d1f26fe root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 6 proof steps

e-1435140a6a62e1e4 basis_map (A013610) → basis_map_A013610 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-2cd9cd27e716475b basis_map (A007318) → basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-621bc4ded5b72353 basis_map (A013611) → basis_map_A013611 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-62dc2e1e9139053e basis_map (A008277) → basis_map_A008277 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-aca51a80a394c762 basis_map (A013609) → basis_map_A013609 of Triangle T(n, k) of numbers of square lattice walks that start and e…
e-510754e51c3ada00 gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and en…

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A008459: Square the entries of Pascal's triangle.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023